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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

CHAPTER 6Frequency Response, Bode

Plots, andResonance

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

CHAPTER 6Frequency Response, Bode

Plots, and Resonance1. State the fundamental concepts of Fourier

analysis.

2. Determine the output of a filter for a given input consisting of sinusoidal components using the filter’s transfer function.

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

3. Use circuit analysis to determine thetransfer functions of simple circuits.

4. Draw first-order lowpass or highpass filter

circuits and sketch their transfer functions.

5. Understand decibels, logarithmic frequency

scales and Bode plots

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

6. Draw the Bode plots for transfer functions of first-order filters.

7. Use software to produce Bode plots for more

complex RLC filters.

8. Calculate parameters for series and parallel resonant circuits.

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

9. Select and design simple filter circuits.

10. Design simple digital signal-processing

systems.

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

Fourier Analysis

All real-world signals are sums of sinusoidal components having various frequencies, amplitudes, and phases.

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

Filters

Filters process the sinusoid components of an inputsignal differently depending of the frequency of each component. Often, the goal of the filter is to retain the components in certain frequency ranges and to reject components in other ranges.

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

Transfer Functions

The transfer function H(f ) of the two-port filter is defined to be the ratio of the phasor output voltage to the phasor input voltage as a function of frequency:

( )in

out

VV

=fH

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

The magnitude of the transfer function shows how the amplitude of each frequency component is affected by the filter. Similarly, the phase of the transfer function shows how the phase of eachfrequency component is affected by the filter.

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

Determining the output of a filter for an input with multiple

components:

1. Determine the frequency and phasor representation for each input component.

2. Determine the (complex) value of the transfer function for each component.

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

3. Obtain the phasor for each output component by multiplying the phasor for each input component by the corresponding transfer-function value.

4. Convert the phasors for the output components into time functions of various frequencies. Add these time functions to produce the output.

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

Linear circuits behave as if they:

1. Separate the input signal into components having various frequencies.

2. Alter the amplitude and phase of each component depending on its frequency.

3. Add the altered components to produce the output signal.

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

FIRST-ORDER LOWPASSFILTERS

RCfB π2

1=

( ) ( )BffjfH

+=

11

( )( )21

1

BfffH

+=

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−=∠

BfffH arctan

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

DECIBELS, THE CASCADE CONNECTION, AND LOGARITHMIC

FREQUENCY SCALES

( ) ( )fHfH log20dB=

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

Cascaded Two-Port Networks( ) ( ) ( )fHfHfH 21 ×=

( ) ( ) ( )dB2dB1dB

fHfHfH +=

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

Logarithmic Frequency Scales

On a logarithmic scale, the variable is multiplied by a given factor for equalincrements of length along the axis.

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

A decade is a range of frequencies for which the ratio of the highest frequency to the lowest is 10.

⎟⎟⎠

⎞⎜⎜⎝

⎛=

1

2log decades ofnumber ff

An octave is a two-to-one change in frequenc

( )( ) ⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

2logloglog octaves ofnumber 12

1

22

ffff

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

BODE PLOTSA Bode plot shows the magnitude of a network function in decibels versus frequency using a logarithmic scale for frequency.

( ) ( )BffjfH

+=

11

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−=

2

dB1log10

BfffH

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

1. A horizontal line at zero for f < fB /10.

2. A sloping line from zero phase at fB /10 to –90° at 10f

3. A horizontal line at –90° for f > 10fB.

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

FIRST-ORDER HIGHPASS FILTERS

( ) ( )( )B

B

ffjffjfH

+==

1in

out

VV

RCfB π2

1=

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

Computer-Generated Bode Plot

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

( ) 5.00321

3 =++

=RRR

RH

( ) ( ) dB 60.520log0dB −==H

( ) 5263.031

3 =+

=∞RR

RH

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

( ) ( ) dB 575.55263.0log20dB −==∞H

( ) ( )[ ]LjRCjRRRfH

ωω ++++=

231

3

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

SERIES RESONANCE

Resonance is a phenomenon that can be observed in mechanical systems andelectrical circuits.

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

LCf

π21

0 =

RLfQs

02π=

CRfQs

021

π=

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−+=

ff

ffjQRfZ ss

0

0

1

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

Series Resonant Circuit as a Bandpass Filter

( )ffffjQss

R

0011

−+=

VV

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

LH ffB −=

sQfB 0=

20BffH +≅

20BffL −≅

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

PARALLEL RESONANCE

( ) ( )fLjfCjRZ p ππ 2121

1−+

=

LCf

π21

0 = LfRQp

02π= CRfQp 02π=

( )ffffjQRZ

pp

001 −+=

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

Ideal Filters

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

Second-Order Lowpass Filter

( ) ( )( )ffffjQ

ffjQfHs

s

00

0

in

out

1 −+−

==VV

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

DIGITAL SIGNAL PROCESSING

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

Conversion of Signals from Analog to Digital Form

If a signal contains no components with frequencies higher than fH, the signal can be exactly reconstructed from its samples, providedthat the sampling rate fs is selected to be more than twice fH.

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ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Third Edition, by Allan R. Hambley, ©2005 Pearson Education, Inc.

Digital Lowpass Filter

( ) ( ) ( ) ( )nxanayny −+−= 11

TTaτ

τ+

=1

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